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European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis

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  • A. H. Nzokem

Abstract

The paper investigates the performance of the European option price when the log asset price follows a rich class of Generalized Tempered Stable (GTS) distribution. The GTS distribution is an alternative to Normal distribution and $\alpha$-stable distribution for modeling asset return and many physical and economic systems. The data used in the option pricing computation comes from fitting the GTS distribution to the underlying S\&P 500 Index return distribution. The Esscher transform method shows that the GTS distribution preserves its structure. The extended Black-Scholes formula and the Generalized Black-Scholes Formula are applied in the study. The 12-point rule Composite Newton-Cotes Quadrature and the Fractional Fast Fourier (FRFT) algorithms were implemented, and they yield the same European option price at two decimal places. Compared to the option price under the GTS distribution, the Black-Scholes (BS) model is underpriced for the Near-The-Money (NTM) and the in-the-money (ITM) options. However, the BS model and GTS European options yield the same option price for the deep out-of-the-money (OTM) and the deep-in-the-money (ITM) options.

Suggested Citation

  • A. H. Nzokem, 2023. "European Option Pricing Under Generalized Tempered Stable Process: Empirical Analysis," Papers 2304.06060, arXiv.org, revised Feb 2025.
    • Handle: RePEc:arx:papers:2304.06060
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    References listed on IDEAS

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    1. Aubain Hilaire Nzokem, 2023. "Pricing European Options under Stochastic Volatility Models: Case of Five-Parameter Variance-Gamma Process," JRFM, MDPI, vol. 16(1), pages 1-28, January.
    2. A. H. Nzokem, 2022. "Pricing European Options under Stochastic Volatility Models: Case of five-Parameter Variance-Gamma Process," Papers 2201.03378, arXiv.org, revised Jan 2023.
    3. Sharif Mozumder & Ghulam Sorwar & Kevin Dowd, 2015. "Revisiting variance gamma pricing: An application to S&P500 index options," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 2(02), pages 1-24.
    4. Küchler, Uwe & Tappe, Stefan, 2013. "Tempered stable distributions and processes," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4256-4293.
    5. A. H. Nzokem & V. T. Montshiwa, 2022. "Fitting Generalized Tempered Stable distribution: Fractional Fourier Transform (FRFT) Approach," Papers 2205.00586, arXiv.org, revised Jun 2022.
    6. Michael Grabchak & Gennady Samorodnitsky, 2010. "Do financial returns have finite or infinite variance? A paradox and an explanation," Quantitative Finance, Taylor & Francis Journals, vol. 10(8), pages 883-893.
    7. A.H. Nzokem, 2021. "SIS Epidemic Model Birth-and-Death Markov Chain Approach," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 10(4), pages 1-10, July.
    8. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    9. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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    Cited by:

    1. A. H. Nzokem, 2023. "Bitcoin versus S&P 500 Index: Return and Risk Analysis," Papers 2310.02436, arXiv.org.
    2. Aubain Nzokem & Daniel Maposa, 2024. "Fitting the Seven-Parameter Generalized Tempered Stable Distribution to Financial Data," JRFM, MDPI, vol. 17(12), pages 1-29, November.
    3. Aubain Nzokem & Daniel Maposa, 2024. "Fitting the seven-parameter Generalized Tempered Stable distribution to the financial data," Papers 2410.19751, arXiv.org, revised Jan 2025.

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